Optimal. Leaf size=292 \[ -\frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right ) (a d (A b (m+1)-a B (m+5)) (b c (m+1)-a d (m+3))-b c (a B (m+1)+A b (3-m)) (a d (m+1)+b (c-c m)))}{8 a^3 b^3 e (m+1)}+\frac{(e x)^{m+1} (b c-a d) \left (c (a B (m+1)+A b (3-m))-d x^2 (A b (m+1)-a B (m+5))\right )}{8 a^2 b^2 e \left (a+b x^2\right )}+\frac{d (e x)^{m+1} (A b (m+1)-a B (m+5)) (b c (m+1)-a d (m+3))}{8 a^2 b^3 e (m+1)}+\frac{\left (c+d x^2\right )^2 (e x)^{m+1} (A b-a B)}{4 a b e \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.408485, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {577, 459, 364} \[ -\frac{(e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right ) (a d (A b (m+1)-a B (m+5)) (b c (m+1)-a d (m+3))-b c (a B (m+1)+A b (3-m)) (a d (m+1)+b (c-c m)))}{8 a^3 b^3 e (m+1)}+\frac{(e x)^{m+1} (b c-a d) \left (c (a B (m+1)+A b (3-m))-d x^2 (A b (m+1)-a B (m+5))\right )}{8 a^2 b^2 e \left (a+b x^2\right )}+\frac{d (e x)^{m+1} (A b (m+1)-a B (m+5)) (b c (m+1)-a d (m+3))}{8 a^2 b^3 e (m+1)}+\frac{\left (c+d x^2\right )^2 (e x)^{m+1} (A b-a B)}{4 a b e \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 577
Rule 459
Rule 364
Rubi steps
\begin{align*} \int \frac{(e x)^m \left (A+B x^2\right ) \left (c+d x^2\right )^2}{\left (a+b x^2\right )^3} \, dx &=\frac{(A b-a B) (e x)^{1+m} \left (c+d x^2\right )^2}{4 a b e \left (a+b x^2\right )^2}-\frac{\int \frac{(e x)^m \left (c+d x^2\right ) \left (-c (A b (3-m)+a B (1+m))+d (A b (1+m)-a B (5+m)) x^2\right )}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac{(A b-a B) (e x)^{1+m} \left (c+d x^2\right )^2}{4 a b e \left (a+b x^2\right )^2}+\frac{(b c-a d) (e x)^{1+m} \left (c (A b (3-m)+a B (1+m))-d (A b (1+m)-a B (5+m)) x^2\right )}{8 a^2 b^2 e \left (a+b x^2\right )}+\frac{\int \frac{(e x)^m \left (c (A b (3-m)+a B (1+m)) (b c (1-m)+a d (1+m))+d (b c (1+m)-a d (3+m)) (A b (1+m)-a B (5+m)) x^2\right )}{a+b x^2} \, dx}{8 a^2 b^2}\\ &=\frac{d (b c (1+m)-a d (3+m)) (A b (1+m)-a B (5+m)) (e x)^{1+m}}{8 a^2 b^3 e (1+m)}+\frac{(A b-a B) (e x)^{1+m} \left (c+d x^2\right )^2}{4 a b e \left (a+b x^2\right )^2}+\frac{(b c-a d) (e x)^{1+m} \left (c (A b (3-m)+a B (1+m))-d (A b (1+m)-a B (5+m)) x^2\right )}{8 a^2 b^2 e \left (a+b x^2\right )}-\frac{\left (\frac{a d (b c (1+m)-a d (3+m)) (A b (1+m)-a B (5+m))}{b}-c (A b (3-m)+a B (1+m)) (a d (1+m)+b (c-c m))\right ) \int \frac{(e x)^m}{a+b x^2} \, dx}{8 a^2 b^2}\\ &=\frac{d (b c (1+m)-a d (3+m)) (A b (1+m)-a B (5+m)) (e x)^{1+m}}{8 a^2 b^3 e (1+m)}+\frac{(A b-a B) (e x)^{1+m} \left (c+d x^2\right )^2}{4 a b e \left (a+b x^2\right )^2}+\frac{(b c-a d) (e x)^{1+m} \left (c (A b (3-m)+a B (1+m))-d (A b (1+m)-a B (5+m)) x^2\right )}{8 a^2 b^2 e \left (a+b x^2\right )}-\frac{\left (\frac{a d (b c (1+m)-a d (3+m)) (A b (1+m)-a B (5+m))}{b}-c (A b (3-m)+a B (1+m)) (a d (1+m)+b (c-c m))\right ) (e x)^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{b x^2}{a}\right )}{8 a^3 b^2 e (1+m)}\\ \end{align*}
Mathematica [A] time = 0.170024, size = 165, normalized size = 0.57 \[ \frac{x (e x)^m \left (\frac{(b c-a d) \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right ) (-3 a B d+2 A b d+b B c)}{a^2}+\frac{(A b-a B) (b c-a d)^2 \, _2F_1\left (3,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a^3}+\frac{d \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right ) (-3 a B d+A b d+2 b B c)}{a}+B d^2\right )}{b^3 (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.068, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex \right ) ^{m} \left ( B{x}^{2}+A \right ) \left ( d{x}^{2}+c \right ) ^{2}}{ \left ( b{x}^{2}+a \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{2} + A\right )}{\left (d x^{2} + c\right )}^{2} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B d^{2} x^{6} +{\left (2 \, B c d + A d^{2}\right )} x^{4} + A c^{2} +{\left (B c^{2} + 2 \, A c d\right )} x^{2}\right )} \left (e x\right )^{m}}{b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x^{2} + A\right )}{\left (d x^{2} + c\right )}^{2} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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